Automated versus Chemically Intuitive Deconvolution of Density Functional Theory (DFT)-Based Gas-Phase Errors in Nitrogen Compounds

Catalysis models involving metal surfaces and gases are regularly based on density functional theory (DFT) calculations at the generalized gradient approximation (GGA). Such models may have large errors in view of the poor DFT-GGA description of gas-phase molecules with multiple bonds. Here, we analyze three correction schemes for the PBE-calculated Gibbs energies of formation of 13 nitrogen compounds. The first scheme is sequential and based on chemical intuition, the second one is an automated optimization based on chemical bonds, and the third one is an automated optimization that capitalizes on the errors found by the first scheme. The mean and maximum absolute errors are brought down close to chemical accuracy by the third approach by correcting the inaccuracies in the NNO and ONO backbones and those in N–O and N–N bonds. This work shows that chemical intuition and automated optimization can be combined to swiftly enhance the predictiveness of DFT-GGA calculations of gases.


S1. Schematics of the nitrogen compounds
shows the ball-and-stick representations, the names and molecular formulas of the 13 nitrogen compounds under study. Figure S1. Ball-and-stick representation of the nitrogen compounds in this study. Blue balls correspond to nitrogen atoms, red balls to oxygen atoms and white balls to hydrogen atoms. Table S1 shows the DFT-calculated zero-point energies (ZPEs) and TS corrections at T = 298.15 K for the 13 compounds under study together with the values for H2, O2, N2, H2O and NH3, which are necessary for the assessment of the errors. Figure S2 shows that the formation energies of N2O are converged at a plane-wave cutoff of 450 eV. The convergence of the ZPEs is illustrated in Table S2.        O N=O N-N O-H  NH2OH  1  0  0  1  NO  0  1  0  0  HNO  0  1  0  0  NO2  1  1  0  0  NO3  2  1  0  0  trans-HNO2  1  1  0  1  cis-HNO2  1  1  0

S4. Optimization procedure
The nonlinear programming (NLP) optimization problem was formulated as a minimization of the objective functions: where abs() is the absolute value of the function, max() is the maximum value of the function and ∆ i − ∆ i is the absolute difference between the experimental formation Gibbs energy and the DFT corrected value for each substance i. A reformulation of the abs() and max() functions was used to avoid the use of non-smooth functions in the optimization. 3 The absolute value was split into the sum of its positive and negative parts, each represented by a positive variable. Thus, the discontinuity in the derivative of the function abs() is converted into additional restrictions to the optimization. The max function was replaced by 13 inequalities (one for each nitrogen compound) such that the feasible space was enlarged but the discontinuity of the function was removed.
The multi-objective optimization problem formulated for AO1 found that the utopia point was within the feasible space. In other words, the minimum possible MAE 6 corresponds to the minimum possible MAX. However, for AO2 the utopia point was not inside the feasible space and a Pareto front was obtained, see Figure S3.

S5. Free and fixed parameter tests and comparison to previous works
In general, it is assumed that O-H bonds are reasonably well described, but this might vary from one family of compounds to the next. AO1 is not biased by design to assume a good description of O-H bonds, but one could easily assume that error to be zero and reoptimize with it as a fixed parameter. In doing so, via AO1 we obtained a slightly larger MAE and nearly the same values for the rest of the free parameters, see Table S5. While this result calls for a thorough parameter sensitivity analysis that escapes our current scope, it suggests that there is a hierarchy of parameters in AO1 in which O-H bonds are not as important other bonds.  In an earlier work 4 we used the number of oxygen atoms (no) as a descriptor to predict the errors in the DFT-energies of 11 gaseous nitrogen compounds split in three groups, namely NOx, N2OX and HNOx. The rationale behind this descriptor is that the progressive addition of oxygen atoms leads to the formation of multiple bonds, which are poorly described by GGAs. 5 Such correction scheme is based on Equation S3 : For the studied dataset, the enthalpy contribution due to the change in temperature corresponds on average to -0.01 eV/atom, as shown in Table S9. The changes for N2, H2 and O2 are zero at all temperatures. Most enthalpic temperature corrections are smaller than -0.10 eV and there is a nearly perfect cancellation of these contributions when considering chemical reactions, such that the change in the Gibbs energies due to temperature changes is negligible. For instance, consider the reaction: 2 + 3 → 2 5 . The change of enthalpy of this reaction from 0 to 298.15 K amounts to −0.10 − (−0.03) − (−0.05) = −0.02 .
While in general we expect errors within ±0.05 eV for the molecules in Table S9, the largest expectable error is 0.11 eV, which is the span of the values in Table S9 (i.e., NO and NH2OH).
In conclusion, our method is based on free energies of formation, but its corrections are to be used for obtaining accurate reaction energies. Discarding enthalpic contributions due to the changes in temperature would typically lead to errors of magnitudes comparable to the MAEs and MAXs of the methods discussed in the main text. Thus, incorporating enthalpic corrections seems facultative at this point. Figure 5 Equation 6 in the main text is: Because the offsets are in the range of -0.06 to -0.81 eV, it is likely that upon correcting O2 and N2 the compounds will be below the parity line in Figure 5.